I want the "Outlook Today \[Mailbox Display Name\]" to reflect the current display name. When I change a user's display name, how can I update the user's profile display name?
Outlook stores user profiles in the registry under the HKEY_CURRENT_USER\SOFTWARE\Microsoft\Windows NT\CurrentVersion\Windows Messaging Subsystem\Profiles subkey. If you're adventurous, you can tweak various settings and see what they do. But be careful: Microsoft doesn't officially support making these kinds of profile changes.
The best way to solve your particular problem is to write a Windows Script Host (WSH) script to do your bidding. Because the string that Outlook displays next to "Outlook Today" is a profile setting, the trick is to find the right profile subkey. The profile names are easy to find, but Outlook uses globally unique identifiers (GUIDs) to identify the subkeys, which makes locating the right attribute difficult. The attribute you need to change is a string value of type REG_SZ whose name (at least for the US English version of Outlook 2002) is 001e3001. Be aware that the display name you want is the one that includes the string "Mailbox " (or its localized equivalent). One way to make sure you're modifying the right subkey under the profile is to look for the string "Mailbox " in \Profiles\ProfileName\guid\001e3001 and for a string value of type REG_SZ named 001e300a with contents "EMSMDB.DLL" in the same GUID.